The Rings of Earth

As most of you are aware, tidal action between the Earth and Moon is both causing the Earth to slow down in its rotation and the Moon to recede from the Earth.

Without going into too much detail, the essential culprit is the fact that the Earth's rotation tends to drag the tidal bulge with it, ahead of the line joining the center of the Earth and the Moon. This causes an off-center pull which tranfers angular momentum from the Earth to the Moon. (And because of the Laws of Celestrial Mechanics, pushes the Moon in into a higher, slower orbit.)

As you've also likely have heard, this will eventually lead to the the Earth rotating at the same rate as the moon revolves; causing the Earth to always face one side to the Moon. (The tidal effect of the Earth on the Moon has already long ago caused the Moon to be in tidal lock with the Earth.)

So far, so good, but what happens then? Is this situation now stable?

Actually, no.

There is another player on the field; the Sun. And like the Moon has its tidal effects on the Earth, so does the Sun. And as the Earth brought the Moon into tidal lock, the Sun is working on bringing the Earth into tidal lock with the Sun.

The result of this is that the Sun will continue to slow the Earth's rotation. As it does so, the Earth will actually start to rotate slower than the moon revolves around it. Again, friction within the Earth's crust will cause the tidal bulge to be out of alignment with the Moon. Only this time, the bulge will lag behind the Moon and tug it backwards rather than forward. This will have the opposite effect of what is happening now, tugging the Moon back into a lower orbit. Lower orbits are faster, and the Moon will also tug on the Earth in an attempt to speed up its rotatation. But the Sun continues to work at slowing the Earth's rotation, and, as a result, the Earth never quite catches up with the Moon, so the Moon will be progressivley be pulled into a lower and lower orbit.

This will continue until the Moon passes inside the Earth's Roche limit. At which point, the Moon will break up and form a ring sytem around the Earth. Once this happens, and the Moon's mass is equally distributed around the Earth, there will no longer be any off-center tidal pull and things will finally settle down.

The fly in the ointment of all this is the fact that it will take several billion years for it to occur, and our sun will expand into a Red giant long before this, most likely encompassing the Earth and Moon, vaporizing both. (Though I've head of some estimates that have the Red giant only expanding out to the orbit of Venus, so maybe there's a chance, if the Earth-Moon system can survive the blow off of mass and formation of the planetary nebula that marks the end of Red giant stage.)

It's a bit of a shame that there may not be enough time for Earth to form such a ring system, it probably would be quite a sight.

Consider this, the entire ring system of Saturn is estimated to have a mass of only 1/1000000 of the Moon, and it is spread over a diameter of about 250,000 miles. The Earth's Roche limit is about 16,000 miles, so its ring system would be maybe on the order of 32,000 miles in diameter. A lot more material spread over a smaller distance, making than a lot denser.

Granted, Saturn's rings are icey thus more reflective than the rocky ring Earth would have, but the extra amount of material, plus the greater proximity to the sun should make up for this, making the Rings of Earth a sight to behold.

Originally posted by Brad_Ad23 Who knows what technologies we will have if we are around billions of years. Perhaps we will be able to swap stars with some other system. Regardless, there's a lot of time in the future.

There have already been studies done which show that it would be possible to move the Earth further from the Sun as it expands, but that's not really the point. I was focusing on the dynamics of the system and its natural outcome without technological interference.

Janus,
Interesting reading!
How long would it take for the moon to completely break down?
I was thinking it could be at least as visually appealing to witness the ‘in-between’ stages as it would only to view the rings.

I was referring more or less to somehow replacing our sun with an identical star from somewhere else. No clue how, but that is billions of years in the future. The dynamics would then have time to occur.

Originally posted by Janus
This will continue until the Moon passes inside the Earth's Roche limit. At which point, the Moon will break up and form a ring sytem around the Earth.
...
It's a bit of a shame that there may not be enough time for Earth to form such a ring system, it probably would be quite a sight.

What...I'm going to miss seeing it again?!? (the moon may have started out as a ring system around the Earth too)

It seems to me there is another possible "stable state" for this system; with the Earth tidaly locked to the Sun and the Moon slowed to an orbit which matches Earth's rotation, so that one Lunar orbit = one Earth day = one Earth year.

Has the math already been done on this, and do we know that this state will not be achieved?

'Course, then ya gotta worry 'bout the tidal effect of the Earth on the Sun, slowing it's rotation and getting further out in the process...[?]

Originally posted by LURCH It seems to me there is another possible "stable state" for this system; with the Earth tidaly locked to the Sun and the Moon slowed to an orbit which matches Earth's rotation, so that one Lunar orbit = one Earth day = one Earth year.

Has the math already been done on this, and do we know that this state will not be achieved?

I once did the calculation for this type of arrangement, and it turns out to be impossible. Before the Moon could come even close to having its period of revolution match that of one Earth year, it would have moved far beyond the Earth's Gravitational Sphere of Influence. That is the maximum distance a satellite can have before the Sun pulls it away into an independant orbit.

This is calculated by

R = D(Mp/Ms)2/5

D = distance from sun to planet
Mp = Mass of planet
Ms = Mass of Sun.

For the Earth, this distance is about 927,000 km, at which distance, the moon would only have a period of 103 days.

In fact, it turns out that no possible combination of Sun/planet/moon can have such an arrangement where the moon orbits the planet in the same time as the planet orbits the sun.

The only exception to this would be the two Lagrange points located about 1.5 million km sunward and anti-sunward of the Earth, and I really wouldn't consider these positions as "orbiting" the Earth.

There are two problems with this scenerio however.

1. Neither of these postions are really stable, objects placed there are easliy nudged out of position.

2. No body that originates in an regular orbit (such as the Moon) can naturally establish themselves there.

Following is the proof that a moon can't have the same orbital period as its planet. (for normal mass distributions where the sun is much more massive than the planet.)

Given that

Rsp is the distance from sun to planet.
Rmp is the distance from moon to planet
Rgi is the radius of the gravitational sphere of influence for the planet.
Ms is the mass of the star.
Mp is the mass of the planet.
Pp is the orbital period of the planet.
Pm is the orbital period of the moon.

Then

Pp = 2[pi][squ](Rsp³/GMs )

and

Pm = 2[pi][squ](Rmp³/GMp )

if Pp = Pm

then

2[pi][squ](Rsp³/GMs ) = 2[pi][squ](Rmp³/GMp )

[squ](Rsp³/GMs ) = [squ](Rmp³/GMp )

Rsp³/GMs = Rmp³/GMp

GMp/GMs = Rmp³/Rsp³

Mp/Ms = Rmp³/Rsp³

(Mp/Ms)1/3 = Rmp/Rsp

Rmp = Rsp(Mp/Ms)1/3

Now
Rgi = Rsp(Mp/Ms)2/5

And since the mass of the star will always be greater than the mass of the planet (Mp/Ms < 1),

Rmp >Rgi

in all cases where Rmp is the distance the moon must have from the planet in order to have the same period as the planet.

As you've also likely have heard, this will eventually lead to the the Earth rotating at the same rate as the moon revolves; causing the Earth to always face one side to the Moon.

Hm, have you seen the dark side of the moon shining on your neighbourhood?

the rest...

Although the original post makes for interesting reading, it's nothing more than speculative fiction, yes? After all, wouldn't consideration of the tidal effects of the sun and moon combined with the earth, be in the territory of the "3 body problem"? In simple terms, it's a Chaos system: you cannot make predictions of any real certainty over any real length of time.

Hm, have you seen the dark side of the moon shining on your neighbourhood?

Yes, I've seen the dark side of the Moon (it's fully visible every new moon, and at least partially visible any time other than a full moon.) And yes, under proper conditions, it can appear to shine slighty from Earth's reflected light.

But what does this have to do with what I said in my post?

Although the original post makes for interesting reading, it's nothing more than speculative fiction, yes? After all, wouldn't consideration of the tidal effects of the sun and moon combined with the earth, be in the territory of the "3 body problem"? In simple terms, it's a Chaos system: you cannot make predictions of any real certainty over any real length of time.

All this means is that one can not predict with any accuracy when the moon will stop receding and start approaching, or when it will break up.

But the general proccess of recesssion, approach, and break up are well within the realm of such prediction.

Originally posted by Janus Yes, I've seen the dark side of the Moon (it's fully visible every new moon, and at least partially visible any time other than a full moon.) And yes, under proper conditions, it can appear to shine slighty from Earth's reflected light.

There may be a dsicrepency in terminology here. In common parlaince, the phrase "dark side of the moon" usually is a misnomer used to reffer to the side that faces away from Earth (kinda "dark" in our understanding of it; not in EM reception).

But what about the tidal effects of the planets on the Sun? After its collapse, the dwarf Sun will still experience pull from the planets, most notably Jupiter. Shouldn't this eventually kick Jupiter out to a wider orbit and slow the Sun down so that the Sun-Jupiter system is also tidaly locked? (With perterbations for the other planets, of course.)

Originally posted by LURCH There may be a dsicrepency in terminology here. In common parlaince, the phrase "dark side of the moon" usually is a misnomer used to reffer to the side that faces away from Earth (kinda "dark" in our understanding of it; not in EM reception).

True, but even if you replace "dark side" with "farside" in the comment made by I, Brian, They still have nothing to do with what I said concerning the Earth eventually presenting onlky one face to the moon.

But what about the tidal effects of the planets on the Sun? After its collapse, the dwarf Sun will still experience pull from the planets, most notably Jupiter. Shouldn't this eventually kick Jupiter out to a wider orbit and slow the Sun down so that the Sun-Jupiter system is also tidaly locked? (With perterbations for the other planets, of course.)

Well, now you're getting into a pretty complicated dynamic . For one thing, the sun doesn't rotate in one piece. I think you'd have to wait until it cooled down to a brown dwarf for it to do so.

Then you have a number of planets, all with different orbital periods, exerting tidal forces on the sun.

At first, this would be no problem, as they all would be working to slow the sun down as they moved outward. (For now, let's just consider the gas giants).

As the sun slows, it will eventually match rotation with Jupiter's period. The outer gas giants, however, will still be trying to slow it down. Now we'll have a situation like the one like the moon, as the other planets try to drag the sun slower, Jupiter tries to keep it matched with it, and as a result, Jupiter will start to spiral in.

You'll have Jupiter feeding anglular momentum to the sun and the Outer gas giants sucking it away as they move further out.

The sun will now be free to match with Saturn, Once it does so, Saturn will follow Jupiter's example, followed by Uranus.

Once Neptune is left by itself, everything else further out (including Pluto and Charon) would probably be evenly distributed enough to have their tidal effects swamped out.

But by now, we're talking a long, long, long, long time into the future. It also assumes that nothing else excitng happens in the meantime. (a large rogue body passing through the solar system for example.)

Yes, I've seen the dark side of the Moon (it's fully visible every new moon, and at least partially visible any time other than a full moon.) And yes, under proper conditions, it can appear to shine slighty from Earth's reflected light.

If you're trying to take me for task over using "dark side" rather than "far side" then you are indeed correct. However, if you're trying to claim that you can see the far side of the moon then you have it completely wrong. The far side of the moon is always away from the earth, as the moon rotates on its axis once for every revolution around the earth, resulting in the same side always facing the earth. This is basic physics.

As to the rest about planets spontaneously disintregrating - great fiction, but you haven't got anything to support your assumptions.

If you're trying to take me for task over using "dark side" rather than "far side" then you are indeed correct. However, if you're trying to claim that you can see the far side of the moon then you have it completely wrong. The far side of the moon is always away from the earth, as the moon rotates on its axis once for every revolution around the earth, resulting in the same side always facing the earth. This is basic physics.

Where have I said otherwise?

In fact, from my first post:"...this will eventually lead to the the Earth rotating at the same rate as the moon revolves; causing the Earth to always face one side to the Moon. (The tidal effect of the Earth on the Moon has already long ago caused the Moon to be in tidal lock with the Earth.)"

The part in parenthesis refers specifically to this fact. That is what tidal locking is.

I don't see how you can in any way get from this quote that I am claiming that we see the farside of the Moon from the Earth. I am merely stating that with time the Earth will become tidally locked with the Moon.

As to the rest about planets spontaneously disintregrating - great fiction, but you haven't got anything to support your assumptions.

Only the law of gravity, the law of conservation of angular momentum and the laws of celestrial mechanics.

Apologies for the misunderstanding on the "far side" of the moon. I misread your initial statement as referring to the moon showing one side to the earth, rather than the moon only visible from one side of the earth.

As for the tidal forces ripping planets apart - my objection isn't the principle, as much as that you imply that it is either common or inevitable. I would be interested to see where your sources are for the moon and Jupiter specifically being destroyed by future tidal forces. This is especially in view of the fact that you are claiming to make predictions about chaos systems where no prediction is possible without an absurd degree of error.

Originally posted by I, Brian Apologies for the misunderstanding on the "far side" of the moon. I misread your initial statement as referring to the moon showing one side to the earth, rather than the moon only visible from one side of the earth.

As for the tidal forces ripping planets apart - my objection isn't the principle, as much as that you imply that it is either common or inevitable. I would be interested to see where your sources are for the moon and Jupiter specifically being destroyed by future tidal forces. This is especially in view of the fact that you are claiming to make predictions about chaos systems where no prediction is possible without an absurd degree of error.

Massive bodies have tidal effects on each other. These tidal effects cause frictional heating and transfers of angular momentum when the bodies move relative to each other. This, in turn, bleeds off energy as heat which is lost to the system of bodies as it is radiated away. With orbital systems, at least some of the bodies must move inward to compensate.

Certainly interesting, but the objection isn't about conversation of energy as much as the inevitability of the scenario postulated. Perhaps I simply approached this thread in the wrong manner. Certainly, the break of planets due to tidal forces is interesting - but without quantified examples it remains an interesting speculation. Certainly, the link to Rings and Resonances makes the particular point that this idea has important consequences to planetary and lunar formation, none of which are supported.

That's assuming that I read it right with the claim that the moon could not have formed from the earth, as it orbits within the Roche limit - which goes against the current thinking - never a bad thing, but it just means something else less to back up the theory.

There are far too many variables and room for error in making the generalisations for planets in general.